# Iterated filtering

Iterated filtering algorithms are a tool for maximum likelihood inference on partially observed dynamical systems. Stochastic perturbations to the unknown parameters are used to explore the parameter space. Applying sequential Monte Carlo (the particle filter) to this extended model results in the selection of the parameter values that are more consistent with the data. Appropriately constructed procedures, iterating with successively diminished perturbations, converge to the maximum likelihood estimate.[1][2] Iterated filtering methods have so far been used most extensively to study the infectious disease transmission dynamics. Case studies include cholera,[3][4] influenza,[5][6][7] malaria,[8][9][10] HIV,[11] pertussis[12][13] and measles.[4][14] Other areas which have been proposed to be suitable for these methods include ecological dynamics[15] and finance.[16]

The perturbations to the parameter space play several different roles. Firstly, they smooth out the likelihood surface, enabling the algorithm to overcome small-scale features of the likelihood during early stages of the global search. Secondly, Monte Carlo variation allows the search to escape from local minima. Thirdly, the iterated filtering update uses the perturbed parameter values to construct an approximation to the derivative of the log likelihood even though this quantity is not typically available in closed form. Fourthly, the parameter perturbations help to overcome numerical difficulties that can arise during sequential Monte Carlo.

## Overview

The data are a time series $y_1,\dots,y_N$ collected at times $t_1 < t_2 < \dots < t_N$. The dynamic system is modeled by a Markov process $X(t)$ which is generated by a function $f(x,s,t,\theta,W)$ in the sense that

$X(t^{}_n)=f(X(t^{}_{n-1}),t^{}_{n-1},t^{}_n,\theta,W) \,$

where $\theta$ is a vector of unknown parameters and $W$ is some random quantity that is drawn independently each time $f(.)$ is evaluated. An initial condition $X(t_0)$ at some time $t_0, together with a measurement density $g(y_n|X_n,t_n,\theta)$ completes the specification of a partially observed Markov process. A basic iterated filtering algorithm is as follows:

## Input

A partially observed Markov model specified as above
Algorithmic parameters: Monte Carlo sample size $J$; number of iterations $M$; cooling parameters $0 and $b$; covariance matrix $\Phi$; initial parameter vector $\theta^{(1)}$

## Procedure: Iterated filtering

for $m^{}_{}=1$ to $M^{}_{}$
set $X_F(t^{}_0,j)=X(t_0)$ for $j=1,\dots, J$
draw $\theta(t^{}_0,j)\sim \mathrm{Normal}(\theta^{(m)},b a^{m-1} \Phi)$
set $\bar\theta(t^{}_0)=\theta^{(m)}$
for $n^{}_{}=1$ to $N^{}_{}$
set $X_P(t^{}_n,j)=f(X_F(t^{}_{n-1},j),t^{}_{n-1},t_n,\theta(t_{n-1},j),W)$
set $w(n,j) = g(y_n|X_P(t^{}_n,j),t^{}_n,\theta(t_{n-1},j))$
draw $k^{}_1,\dots,k^{}_J$ such that $P(k^{}_j=i)=w(n,i)\big/{\sum}_\ell w(n,\ell)$
set $X_F(t^{}_n,j)=X_P(t^{}_n,k^{}_j)$
draw $\theta(t^{}_n,j)\sim \mathrm{Normal}(\theta(t^{}_{n-1},k^{}_j), a^{m-1} \Phi)$
set $\bar\theta_i^{}(t_n^{})$ to the sample mean of $\{\theta^{}_i(t^{}_{n-1},k^{}_j),j=1,\dots,J\}$, where $\theta$ has components $\{\theta^{}_i\}$
set $V_i^{}(t_n^{})$ to the sample variance of $\{\theta_i^{}(t^{}_{n},k^{}_j),j=1,\dots,J\}$
set $\theta_i^{(m+1)}= \theta_i^{(m)}+V_i(t_{1})\sum_{n=1}^N V_i^{-1}(t_{n})(\bar\theta_i(t_n)-\bar\theta_i(t_{n-1}))$

## Output

maximum likelihood estimate $\hat\theta=\theta^{(M+1)}$

## Variations

1. If $X(t_0)$ is unknown, it can be included in $\theta$. However, such parameters warrant some special algorithmic attention since information about them in the data may be concentrated in a small part of the time series.[1]
2. Theoretically, any distribution with the requisite mean and variance could be used in place of the normal distribution. It is standard to use the normal distribution and to reparameterise to remove constraints on the possible values of the parameters.
3. Modifications to the algorithm have been proposed to give superior asymptotic performance.[17][18][19]

## References

1. ^ a b Ionides, E. L.; Breto, C. and King, A. A. (2006). "Inference for nonlinear dynamical systems". Proceedings of the National Academy of Sciences of the USA 103 (49): 18438–18443. doi:10.1073/pnas.0603181103. PMC 3020138. PMID 17121996.
2. ^ Ionides, E. L.; Bhadra, A., Atchade, Y. and King, A. A. (2011). "Iterated filtering". Annals of Statistics 39 (3): 1776–1802. doi:10.1214/11-AOS886.
3. ^ King, A. A.; Ionides, E. L., Pascual, M. and Bouma, M. J. (2008). "Inapparent infections and cholera dynamics". Nature 454 (7206): 877–880. doi:10.1038/nature07084. PMID 18704085.
4. ^ a b Breto, C.; He, D., Ionides, E. L. and King, A. A. (2009). "Time series analysis via mechanistic models". Annals of Applied Statistics 3: 319–348. doi:10.1214/08-AOAS201.
5. ^ He, D.; J. Dushoff, T. Day, J. Ma and D. Earn (2011). "Mechanistic modelling of the three waves of the 1918 influenza pandemic". Theoretical Ecology 4 (2): 1–6. doi:10.1007/s12080-011-0123-3.
6. ^ Camacho, A.; S. Ballesteros, A. L. Graham, R. Carrat, O. Ratmann and B. Cazelles (2011). "Explaining rapid reinfections in multiple-wave influenza outbreaks: Tristan da Cunha 1971 epidemic as a case study". Proceedings of the Royal Society B 278 (1725): 3635–3643. doi:10.1098/rspb.2011.0300.
7. ^ Earn, D.; He, D., Loeb, M. B., Fonseca, K., Lee, B. E. and Dushoff, J. (2012). "Effects of School Closure on Incidence of Pandemic Influenza in Alberta, Canada". Annals of Internal Medicine 156: 173–181. doi:10.1059/0003-4819-156-3-201202070-00005.
8. ^ Laneri, K.; A. Bhadra, E. L. Ionides, M. Bouma, R. C. Dhiman, R. S. Yadav and M. Pascual (2010). "Forcing versus feedback: Epidemic malaria and monsoon rains in NW India". PLoS Computational Biology 6 (9): e1000898. doi:10.1371/journal.pcbi.1000898. PMC 2932675. PMID 20824122.
9. ^ Bhadra, A.; E. L. Ionides, K. Laneri, M. Bouma, R. C. Dhiman and M. Pascual (2011). "Malaria in Northwest India: Data analysis via partially observed stochastic differential equation models driven by Lévy noise". Journal of the American Statistical Association 106 (494): 440. doi:10.1198/jasa.2011.ap10323.
10. ^ Roy, M.; Bouma, M. J., Ionides, E. L., Dhiman, R. C. and Pascual, M. (2013). "The potential elimination of Plasmodium vivax malaria by relapse treatment: Insights from a transmission model and surveillance data from NW India". PLoS Neglected Tropical Diseases 7: e1979. doi:10.1371/journal.pntd.0001979.
11. ^ Zhou, J.; Han, L. and Liu, S. (2013). Statistics and Probability Letters 83: 1448–1456. doi:10.1016/j.spl.2013.01.032. Missing or empty |title= (help)
12. ^ Lavine, J.; Rohani, P. (2012). "Resolving pertussis immunity and vaccine effectiveness using incidence time series". Expert Review of Vaccines 11: 1319–1329. doi:10.1586/ERV.12.109.
13. ^ Blackwood, J.; Cummings, D. A. T., Broutin, H., Iamsirithaworn, S. and Rohani, P. (2013). "Deciphering the impacts of vaccination and immunity on pertussis epidemiology in Thailand". Proceedings of the National Academy of Sciences of the USA. prepublished online. doi:10.1073/pnas.1220908110.
14. ^ He, D.; Ionides,E. L. and King, A. A. (2010). "Plug-and-play inference for disease dynamics: measles in large and small towns as a case study". Journal of the Royal Society Interface 7 (43): 271–283. doi:10.1098/rsif.2009.0151. PMC 2842609. PMID 19535416.
15. ^ Ionides, E. L.. (2011). "Discussion on "Feature Matching in Time Series Modeling" by Y. Xia and H. Tong.". Statistical Science. doi:10.1214/11-STS345C.
16. ^ Bhadra, A. (2010). "Discussion of "Particle Markov chain Monte Carlo methods" by C. Andrieu, A. Doucet and R. Holenstein". Journal of the Royal Statistical Society, Series B 72 (3): 314–315. doi:10.1111/j.1467-9868.2009.00736.x.
17. ^ Lindstrom, E.; Ionides, E. L., Frydendall, J. and Madsen, H. (2012). "Efficient Iterated Filtering". System Identification 16: 1785–1790. doi:10.3182/20120711-3-BE-2027.00300.
18. ^ Lindstrom, E. (2013). "Tuned iterated filtering". Statistics and Probability Letters 83 (9): 2077––2080. doi:10.1016/j.spl.2013.05.019.
19. ^ Doucet, A.; Jacob, P. E. and Rubenthaler, S. (2013). "Derivative-Free Estimation of the Score Vector and Observed Information Matrix with Application to State-Space Models". arXiv:1304.5768 [stat.ME].